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gaussCol -- Makes column reductions of a matrix

Synopsis

Description

This function does certain column reductions in a matrix M, if M is of a certain type as in this example.

i1 : S=ZZ[x,y];
 
i2 : R=S[a_1..a_4];
 
i3 : M=matrix{{1,a_2,a_3,a_4}};

             1      4
o3 : Matrix R  <-- R
 
i4 : K=nextDegree(ker M,1,S)

o4 = {-1, 0} | a_2 0   a_3 0   a_4 0   |
     {-1, 0} | -1  a_2 0   a_3 0   a_4 |
     {0, 0}  | 0   -1  0   0   0   0   |
     {0, 0}  | 0   0   -1  0   0   0   |
     {0, 0}  | 0   0   0   -1  -1  0   |
     {0, 0}  | 0   0   0   0   0   -1  |

             6      6
o4 : Matrix R  <-- R

What it does is that it starts from the bottom right and if it finds a -1 then it uses that column to reduces all other columns to a form with a zero in the corresponding row. It then continues up left searching for the next -1 and repeats the procedure.

i5 : gaussCol(K)

o5 = | a_2 a_2^2 a_3 a_2a_3 -a_2a_3+a_4 a_2a_4 |
     | -1  0     0   0      0           0      |
     | 0   -1    0   0      0           0      |
     | 0   0     -1  0      0           0      |
     | 0   0     0   -1     0           0      |
     | 0   0     0   0      0           -1     |

             6      6
o5 : Matrix R  <-- R

See also

Ways to use gaussCol :

For the programmer

The object gaussCol is a method function.